The role of X-ray based electron spectroscopies in determining chemical, electronic and magnetic properties of solids has been well known for several decades. A powerful approach is angle-resolved photoelectron spectroscopy, whereby the kinetic energy and angle of photoelectrons emitted from a sample surface are measured. This provides a direct measurement of the electronic band structure of crystalline solids. Moreover, it yields powerful insights into the electronic interactions at play within a material, and into the control of spin, charge and orbital degrees of freedom; central pillars of future solid state science. With strong recent focus on research of lower-dimensional materials and modified electronic behaviour at surfaces and interfaces, angle-resolved photoelectron spectroscopy has become a core technique in the study of quantum ma-1 terials. In this review, we provide an introduction to the technique. Through examples from several topical materials systems, including topological insulators, transition metal dichalcogenides, and transition metal oxides, we highlight the types of information which can be obtained. We show how the combination of angle, spin, time and depth-resolved experiments are able to reveal 'hidden' spectral features, connected to semiconducting, metallic and magnetic properties of solids, as well as underlining the importance of dimensional effects in quantum materials.